Simplify and expand the following expression: $ \dfrac{3}{a - 4}- \dfrac{4}{5a - 25}- \dfrac{2a}{a^2 - 9a + 20} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{4}{5a - 25} = \dfrac{4}{5(a - 5)}$ We can factor the quadratic in the third term: $ \dfrac{2a}{a^2 - 9a + 20} = \dfrac{2a}{(a - 4)(a - 5)}$ Now we have: $ \dfrac{3}{a - 4}- \dfrac{4}{5(a - 5)}- \dfrac{2a}{(a - 4)(a - 5)} $ The least common multiple of the denominators is: $ (a - 4)(a - 5)$ In order to get the first term over $(a - 4)(a - 5)$ , multiply by $\dfrac{5(a - 5)}{5(a - 5)}$ $ \dfrac{3}{a - 4} \times \dfrac{5(a - 5)}{5(a - 5)} = \dfrac{15(a - 5)}{(a - 4)(a - 5)} $ In order to get the second term over $(a - 4)(a - 5)$ , multiply by $\dfrac{a - 4}{a - 4}$ $ \dfrac{4}{5(a - 5)} \times \dfrac{a - 4}{a - 4} = \dfrac{4(a - 4)}{(a - 4)(a - 5)} $ In order to get the third term over $(a - 4)(a - 5)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{2a}{(a - 4)(a - 5)} \times \dfrac{5}{5} = \dfrac{10a}{(a - 4)(a - 5)} $ Now we have: $ \dfrac{15(a - 5)}{(a - 4)(a - 5)} - \dfrac{4(a - 4)}{(a - 4)(a - 5)} - \dfrac{10a}{(a - 4)(a - 5)} $ $ = \dfrac{ 15(a - 5) - 4(a - 4) - 10a} {(a - 4)(a - 5)} $ Expand: $ = \dfrac{15a - 75 - 4a + 16 - 10a}{5a^2 - 45a + 100} $ $ = \dfrac{a - 59}{5a^2 - 45a + 100}$